System of Equations Calculator with Steps for 2 and 3 Variables
Our system of equations calculator solves two or three linear equations and shows every step. It works as a substitution calculator, an elimination tool and a determinant solver in one place. Fractions and decimals work, and the answer is exact.
How to Use the System of Equations Calculator
Choose two equations with two unknowns or three equations with three unknowns. Type the coefficients of x, y and z in each row, then the number on the right side of the equals sign. An empty box counts as 0, so a missing term needs no typing.
Pick a method for the steps. The answer is the same for all three methods, but the working looks different. The system of equations calculator also draws both lines when you solve a pair of equations.
Fractions such as 3/4 and decimals such as 0.75 are exact inside the tool. Nothing is rounded until the final display. Use it as a substitution calculator when you want to compare each rewritten equation with your own working.
Three Ways to Solve a System
Elimination Method
The elimination method removes one unknown at a time. You add or subtract multiples of one equation from another until a single unknown is left. The tool does this on an augmented matrix, which lists each equation as a row. It is a good choice for a 3 by 3 system of equations because the steps stay short.
Substitution Method
The substitution method solves one equation for one unknown and puts the result into the others. Our substitution calculator picks an equation with a coefficient of 1 or −1 when it can, because that avoids fractions. It repeats the idea until one unknown remains and then works back up.
Cramer's Rule
This rule finds each unknown as a ratio of determinants. D is the determinant of the coefficients, and Dx swaps the x column for the right side. The rule needs D to be different from 0. When D is 0 the system has no single answer, and the tool switches to row reduction to explain why.
Reading the Result
A system can have one solution, no solution or infinitely many solutions. One solution means the lines or planes cross at a single point. No solution means they never meet, as with parallel lines. Infinitely many solutions means the equations describe the same line or the same plane.
The determinant D appears beside the answer. A value of 0 signals that the system is not a single point. Simultaneous equations like these come up in algebra courses and in physics, finance and engineering problems.
A Worked Example with Two Unknowns
Take 2x + 3y = 13 and x − y = −1. These values are already filled in, so the page shows the same working when it opens. The determinant is D = 2 × (−1) − 3 × 1, which is −5.
With elimination, the tool divides row 1 by 2 and then subtracts row 1 from row 2. It divides row 2 by −5/2 and clears the first row. The result is x = 2 and y = 3.
Cramer's rule agrees. Dx = 13 × (−1) − 3 × (−1) = −10 and Dy = 2 × (−1) − 13 × 1 = −15. Then x = −10 ÷ −5 = 2 and y = −15 ÷ −5 = 3.
Our substitution calculator gives the same answer. From x − y = −1 you get x = y − 1. Put that into the first equation to get 2(y − 1) + 3y = 13, so 5y = 15 and y = 3. Then x = 2.
A Worked Example with Three Unknowns
Switch the first box to three equations to try this 3 by 3 system of equations: x + y + z = 6, 2y + 5z = −4 and 2x + 5y − z = 27. The solution is x = 5, y = 3 and z = −2. The first equation checks as 5 + 3 − 2 = 6. The second gives 6 − 10 = −4, and the third gives 10 + 15 + 2 = 27.
The tool lists every row operation and the matrix after it, so you can follow along on paper. It ends with a check that puts the answer back into each equation. That check is the quickest way to catch a typing error.
When There Is No Solution or Many Solutions
Try x + y = 2 and x + y = 5. The same sum cannot equal two numbers, so there is no solution. The graph shows two parallel lines.
Now try x + y = 2 and 2x + 2y = 4. The second equation is the first one doubled, so every point on the line works. The system of equations calculator then writes x in terms of a free variable y.
Why Use a Substitution Calculator
A substitution calculator helps most when you are learning. Solve a problem by hand and then compare each line with the tool. If your working differs, you will see exactly where the first slip happened. The system of equations calculator also keeps every fraction exact, which avoids rounding errors.
Teachers often ask for a specific method, so the method selector matters. Choose substitution for a worked answer that matches class notes. Choose elimination when you need speed, and the determinant method when the question asks for it.
Real Problems You Can Model
A system of equations calculator is useful outside the classroom. Two ticket prices and two totals give a pair of equations for the number of adult and child tickets. Mixing two solutions to reach a target strength gives another pair. Simultaneous equations of this type also appear in budgeting, where two plans share one total.
Three unknowns show up in nutrition plans, circuit analysis and supply problems. Write one equation for each condition and keep the units the same across rows. A substitution calculator handles these small systems well, and it returns the single set of numbers that satisfies all of them.
Tips for Entering Equations
Rewrite each equation so the unknowns sit on the left and the constant sits on the right. Move 3x + 2 = y to 3x − y = −2 before you type it. Keep the order x, y and z in every row.
A substitution calculator needs one clear equation to start from, so tidy input gives shorter working. If an equation has brackets, expand them first. For decimals, type them as they are, and the tool converts them to exact fractions.
The system of equations calculator cannot read words or powers of the unknowns. It solves linear systems only, so terms such as x squared need a different tool, like the quadratic solver.
Common Mistakes
- Typing a coefficient in the wrong column, such as a y term under x.
- Dropping a minus sign when you move a term across the equals sign.
- Leaving a term out and forgetting that an empty box means 0.
Read your system back from the Steps panel first. It prints the equations in plain form before it solves them. If that line does not match your homework, fix the input before you trust the answer.
Frequently asked questions
Is this system of equations calculator free?
Yes, it is free and runs in your browser.
Can it solve three equations?
Yes, choose three equations and type the nine coefficients and three constants.
How does the substitution calculator work?
It solves one equation for one unknown, substitutes it and repeats.
What if the determinant is 0?
The system has no solution or infinitely many, and the steps show which one.
Does it accept fractions?
Yes, type 3/4 or 0.75 and the working stays exact.
Is the substitution calculator accurate?
Yes, it uses exact fractions, so nothing is rounded until the final display.
Can the system of equations calculator handle three unknowns?
Yes, it solves for one unknown, then repeats until one remains.
Can I see the graph?
Yes, the two line graph appears when you solve two equations.
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